Large language models (LLMs) are being used in many applications and prompts for these models are integrated into software applications as code-like artifacts. These prompts behave much like traditional software in that they take inputs, generate outputs, and perform some specific function. However, prompts differ from traditional code in many ways and require new approaches to ensure that they are robust. For example, unlike traditional software the output of a prompt depends on the AI model that interprets it. Also, while natural language prompts are easy to modify, the impact of updates is harder to predict. New approaches to testing, debugging, and modifying prompts with respect to the model running them are required. To address some of these issues, we developed PromptPex, an LLM-based tool to automatically generate and evaluate unit tests for a given prompt. PromptPex extracts input and output specifications from a prompt and uses them to generate diverse, targeted, and valid unit tests. These tests are instrumental in identifying regressions when a prompt is changed and also serve as a tool to understand how prompts are interpreted by different models. We use PromptPex to generate tests for eight benchmark prompts and evaluate the quality of the generated tests by seeing if they can cause each of four diverse models to produce invalid output. PromptPex consistently creates tests that result in more invalid model outputs than a carefully constructed baseline LLM-based test generator. Furthermore, by extracting concrete specifications from the input prompt, PromptPex allows prompt writers to clearly understand and test specific aspects of their prompts. The source code of PromptPex is available at https://github.com/microsoft/promptpex.